Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x-9y &= 5 \\ -2x-5y &= 5\end{align*}$
Solution: Begin by moving the $y$ -term in the second equation to the right side of the equation. $-2x = 5y+5$ Divide both sides by $-2$ to isolate $x$ $x = {-\dfrac{5}{2}y - \dfrac{5}{2}}$ Substitute this expression for $x$ in the first equation. $2({-\dfrac{5}{2}y - \dfrac{5}{2}}) - 9y = 5$ $-5y - 5 - 9y = 5$ Simplify by combining terms, then solve for $y$ $-14y - 5 = 5$ $-14y = 10$ $y = -\dfrac{5}{7}$ Substitute $-\dfrac{5}{7}$ for $y$ in the top equation. $2x-9( -\dfrac{5}{7}) = 5$ $2x+\dfrac{45}{7} = 5$ $2x = -\dfrac{10}{7}$ $x = -\dfrac{5}{7}$ The solution is $\enspace x = -\dfrac{5}{7}, \enspace y = -\dfrac{5}{7}$.